I'm an auto mechanic & I'm presuming you are talking about automotive oil. How much hit the camera? A small amount might penetrate the body but I doubt it would harm the inside as much as maybe the rubbery grip stuff on the outside. As for the LCD there is a protective cover over the actual LCD and unless the oil gets between them & softens your view of the images on the LCD. As for a clean up you're really only going to be able to get oil on the outside at this point & I'd do it with dry methods (paper towels or the blue shop towels that are like paper towels etc.
I'm an auto mechanic & I'm presuming you are talking about automotive oil.
That's funny, shows how different all our minds are based on our backgrounds. I'm an engineer, and I pictured it as a light oil like a mineral oil such as sewing machine oil or for a bike chain (well, now I use teflon for that). I wonder if a chef would have imagined a cooking oil.
I would probably be more worried if it was a thicker oil. But a little light oil shouldn't gum up things too much. In fact usually you'd use a light oil to keep the seals functioning smoothly.
If it was me, I wouldn't spend the money sending an older camera in like a 7D. I might be tempted with a new camera, but even then, I think I'd just wipe it down with soapy water and let it dry a bit. If I was really worried I'd probably pull it apart and clean it up a bit inside.